Question: All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$4.50$ each for students, and the group paid $$37.50$ in total. The next month, the same group visited a science museum where the tickets cost $$10.00$ each for teachers and $$8.50$ each for students, and the group paid $$72.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+4.5y = 37.5}$ ${10x+8.5y = 72.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-10x-9y = -75}$ ${10x+8.5y = 72.5}$ Add the top and bottom equations together. $ -0.5y = -2.5 $ $ y = \dfrac{-2.5}{-0.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {5x+4.5y = 37.5}$ to find $x$ ${5x + 4.5}{(5)}{= 37.5}$ $5x+22.5 = 37.5$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {10x+8.5y = 72.5}$ and get the same answer for $x$ ${10x + 8.5}{(5)}{= 72.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.